I'm not sure we're talking about the same situation.
But: in a standard right turn at a 4-way intersection in California, where neither street is one-way, when turning right, you must turn into the rightmost lane of the cross street. In particular (barring special things like dual right-turn lanes), you can't swing wide into the second lane of the cross street. By "second lane," I don't mean the lane for opposing traffic (obviously you can't go into that lane) -- I mean the left lane of the two that are going in the direction you are turning.
If you are making a left turn, you can go into either the left lane or the right lane of the cross street.
